Generalized periodicity in group cohomology

Abstract

Given a finite group G, we introduce “encoding pairs,” which are a pair of G-modules M and $M\prime$ equipped with a shifted natural isomorphism between the cohomological functors $H^{•}(G,{\text{Hom}}_{\mathbb{Z}}(M,-))$ and $H^{•}(G,{\text{Hom}}_{\mathbb{Z}}(M\prime ,-))$. Studying these encoding pairs generalizes the theory of periodic cohomology for finite groups, allowing us to generalize the cohomological input of a theorem due to Swan that roughly says that a finite group with periodic cohomology acts feely on some sphere.

KEYWORDS:

• Finite group
• periodic cohomology

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20J06

Acknowledgments

This research was conducted at the University of Michigan REU during the summer of 2022. The author would especially like to thank his advisors Alexander Bertoloni Meli, Patrick Daniels, and Peter Dillery for their eternal patience and guidance, in addition to a number of helpful comments on earlier drafts of this article. Without their advice, this project would have been impossible. The author would also like to thank Maxwell Ye for a number of helpful conversations and consistent companionship. Without him, the author would have been left floating adrift and soulless. Additionally, the author would like to thank the anonymous referee for a careful reading and many detailed comments on an earlier draft.

Additional information

Funding

It was supported by NSF DMS-1840234.